Direct and Inverse Theorems in the Theory ofApproximation of Banach Space Vectors by Ex-ponential Type Entire Vectors

Yaroslav Grushka and Sergiy Torba

Abstract. An arbitrary operator A on a Banach space X such that eitherA or iA generates the C0–group with certain growth condition at infinity isconsidered. The direct and inverse theorems on connection between the degreeof smoothness of a vector x ∈ X with respect to the operator A, the rate ofconvergence to zero of the best approximation of x by exponential type entirevectors for the operator A, and the k-module of continuity are established.These results allow to obtain Jackson-type and Bernstein-type inequalities inweighted Lp spaces.

Mathematics Subject Classification (2000). Primary 41A25, 41A27, 41A17,41A65.

Keywords. Direct and inverse theorems, Module of continuity, Banach space,Non-quasianalytic operators, Entire vectors of exponential type.

1. Introduction

Direct and inverse theorems which establish the relationship between the degree ofsmoothness of a function with respect to the differentiation operator and the rateof convergence to zero of its best approximation by trigonometric polynomials arewell known in the theory of approximation of periodic functions. Jackson’s andBernstein’s inequalities are ones among such results.

N. P. Kuptsov proposed a generalized notion of module of continuity, ex-panded onto C0-groups in a Banach space [1]. Using this notion, N. P. Kuptsov [1]and A. P. Terekhin [2] proved the generalized Jackson’s inequalities for the cases ofbounded group and s-regular group. Recall that group {U(t)}t∈R is called s-regular

This work was partially supported by the Ukrainian State Foundation for Fundamental Research(project N14.1/003).

2 Ya. Grushka and S. Torba

if resolvent of its generator A satisfies the condition ∃θ ∈ R : ‖Rλ(eiθAs)‖ ≤C

Imλ .G. V. Radzievsky studied direct and inverse theorems [3, 4], using the notion

of K-functional instead of module of continuity, but it should be noted that theK-functional has two-sided estimates with regard to the module of continuity atleast for bounded C0-groups.

In the papers [5, 6] and [7] authors investigated the case of a group of unitaryoperators in a Hilbert space and established Jackson-type inequalities in Hilbertspaces and their rigs. These inequalities are used to estimate the rate of conver-gence to zero of the best approximation of both finite and infinite smoothnessvectors for operator A by exponential type entire vectors.

We consider C0-groups, generated by so-called non-quasianalytic operators[8], i.e. groups satisfying ∫ ∞

−∞

ln ‖U(t)‖1 + t2

dt <∞. (1.1)

We recall that the belonging of group to the C0 class means that for every x ∈ Xvector-function U(t)x is continuous on R with respect to the norm of space X.

As it was shown in [5], the set of exponential type entire vectors for thenon-quasianalytic operator A is dense in X, so the problem of approximation byexponential type entire vectors is correct. On the other hand, it was shown in [9]that condition (1.1) is close to the necessary one, so in the case when (1.1) doesn’thold, the class of entire vectors isn’t necessary dense in X, and the correspondingapproximation problem loses its meaning.

The purpose of this work is to obtain Jackson- and Bernstein-type inequalitiesand inverse theorems in the case where a vector of a Banach space is approximatedby exponential type entire vectors for a non-quasianalytic operator, and to givesome applications of these results to weighted Lp spaces. First, in the section 2,all necessary definitions and the statement of approximation problem are given,next, in the section 3 we describe required apparatus, in the section 4 direct andinverse theorems are established, and, in the last section 5, we give some examplesof using the results in weighted Lp spaces.

2. Preliminaries

Let A be a closed linear operator with dense domain of definition D(A) in theBanach space (X, ‖·‖) over the field of complex numbers.

Let C∞(A) denotes the set of all infinitely differentiable vectors of operatorA, i.e.

C∞(A) =⋂n∈N0

D(An), N0 = N ∪ {0}.

For number α > 0 we set

Eα(A) ={x ∈ C∞(A) | ∃c = c(x) > 0 ∀k ∈ N0

∥∥Akx∥∥ ≤ cαk} .

Direct and Inverse Theorems in the Theory of Approximation ... 3

The set Eα(A) is a Banach space with respect to the norm

‖x‖Eα(A) = supn∈N0

‖Anx‖αn

.

Then E(A) =⋃α>0 Eα(A) is a linear locally convex space with respect to the

topology of inductive limit of Banach spaces Eα(A):

E(A) = lim indα→∞

Eα(A).

Elements of space E(A) are called exponential type entire vectors of operator A.The type σ(x,A) of vector x ∈ E(A) is defined as the number

σ(x,A) = inf {α > 0 : x ∈ Eα(A)} = lim supn→∞

‖Anx‖1n . (2.1)

Denote by Ξα(A) the following set

Ξα(A) ={x ∈ E(A) |σ(x) ≤ α

}. (2.2)

It is easy to see that

Eα(A) ⊂ Ξα(A) =⋂ε>0

Eα+ε(A). (2.3)

The question if the inclusion in the equation (2.3) is strict naturally arises. If theoperator A is a generator of bounded C0-group U(t), then, as follows from [2,Theorem 1], Eα(A) = Ξα(A). But if the group U(t) is unbounded (as is for non-quasianalytic oparators A), it is possible that Eα(A) 6= Ξα(A), that is there existssuch vector x that σ(x) = r, but x 6∈ Er(A). We will return to this question later,discussing the generalization of the Bernstein inequality in the section 4 and inthe section 5, where the example of such operator A and vector x is constructed.

For arbitrary x ∈ X we set, according to [7, 6],

Er(x,A) = infy∈Ξr(A)

‖x− y‖ , r > 0,

i.e. Er(x,A) is the best approximation of element x by exponential type entirevectors y of operator A for which σ(y,A) ≤ r. For fixed x, Er(x,A) does notincrease and Er(x,A) → 0, r → ∞ for every x ∈ X if and only if the set E(A)of exponential type entire vectors is dense in X. Particularly, as indicated above,the set E(A) is dense in X if A generates C0-group {U(t) : t ∈ R} and this groupbelongs to non-quasianalytic class (that is, it satisfies (1.1)).

Example. Let X is one of the Lp(2π) (1 ≤ p < ∞) spaces of integrable in p-thdegree over [0, 2π], 2π-periodical functions or the space C(2π) of continuous 2π-periodical functions (the norm in X is defined in a standard way), and let A isthe differentiation operator in the space X (D(A) = {x ∈ X ∩ AC(R) : x′ ∈ X};(Ax)(t) = dx

dt , where AC(R) denotes the space of absolutely continuous functionsover R). It can be proved that in such case E(A) coincides with the set of alltrigonometric polynomials, and for y ∈ E(A) σ(y,A) = deg(y), where deg(y) isthe degree of trigonometric polynomial y.

4 Ya. Grushka and S. Torba

Note that all previous definitions do not change if we replace the operatorA by any operator of form eiθA, θ ∈ R. Moreover, main results of this article donot depend on which operator generates group U(t) — either A or iA. So, in whatfollows, we always assume that the operator iA is the generator of C0-group oflinear continuous operators {U(t) : t ∈ R} on X. Moreover, we suppose that theoperator A is non-quasianalytic.

For t ∈ R+, we set

MU (t) := supτ∈R, |τ |≤t

‖U(τ)‖ . (2.4)

The estimation ‖U(t)‖ ≤ Meωt for some M,ω ∈ R implies MU (t) < ∞ (for allt ∈ R+). It is easy to see that the function MU (·) has the following properties:

1) MU (t) ≥ 1, t ∈ R+;2) MU (·) is monotonically non-decreasing on R+;3) MU (t1 + t2) ≤MU (t1)MU (t2), t1, t2 ∈ R+.

According to [1], for x ∈ X, t ∈ R+ and k ∈ N we set as a generalization ofmodule of smoothness,

ωk(t, x,A) = sup0≤τ≤t

∥∥∆kτx∥∥ , where (2.5)

∆kh = (U(h)− I)k =

k∑j=0

(−1)k−j(j

k

)U(jh), k ∈ N0, h ∈ R (∆0

h ≡ 1). (2.6)

Moreover, letωk(t, x,A) = sup

|τ |≤t

∥∥∆kτx∥∥ . (2.7)

Remark 2.1. It is easy to see that in the case of bounded group {U(t)} (‖U(t)‖ ≤M, t ∈ R) the quantities ωk(t, x,A) and ωk(t, x,A) are equivalent within constantfactor (ωk(t, x,A) ≤ ωk(t, x,A) ≤ M ωk(t, x,A)), and in the case of isometricgroup (‖U(t)‖ ≡ 1, t ∈ R) these quantities coincide.

It is immediate from the definition of ωk(t, x,A) that for k ∈ N:1) ωk(0, x, A) = 0;2) for fixed x the function ωk(t, x,A) is non-decreasing and is continuous by the

variable t on R+;3) ωk(nt, x,A) ≤

(1 + (n− 1)MU ((n− 1)t)

)kωk(t, x,A) (n ∈ N, t > 0);

4) ωk(µt, x,A) ≤(1 + µMU (µt)

)kωk(t, x,A) (µ, t > 0);

5) for fixed t ∈ R+ the function ωk(t, x,A) is continuous in x.

3. Integration kernels and spectral subspaces

In contrast to the Hilbert space case and selfadjoint operators, we do not havespectral decomposition of operator as the convenient tool to deal with entire vec-tors of exponential type, we need another instruments. This section devoted to two

Direct and Inverse Theorems in the Theory of Approximation ... 5

such things — the investigation of properties of generalizations of classic integralkernels, such as

12π

(sin x

2x2

)2

and1Kn

(sin x

2nx2n

)2n

, (3.1)

and to determining the relationship between entire vectors of exponential type andspectral subspaces of non-quasianalytic operators, constructed in [8].

3.1. One class of integration kernels

Integral kernels, generalizing (3.1), were first constructed in [10]. They are entirefunctions, bounded and fast decreasing on real axis. They will be used later forconstructing of approximation. Now we describe how these kernels are constructedand state their properties.

In what follows we denote as Q the class of functions α : R 7→ R, satisfyingthe following conditions:

I) α(·) is measurable and bounded on any segment [−T, T ] ⊂ R.II) α(t) > 0, t ∈ R.

III) α(t1 + t2) ≤ α(t1)α(t2), t1, t2 ∈ R.IV)

∫∞−∞

|ln(α(t))|1+t2 dt <∞.

Note that these conditions are properties of norm of non-quasianalytic group.Without lost of generality we may assume that the function α(t) satisfies additionalconditions:

V) α(t) ≥ 1, t ∈ R; 1

VI) α(t) is even on R and is monotonically increasing on R+;VII)

∥∥α−1∥∥L1(R)

=∫∞−∞ |α

−1(t)|dt <∞.

Really, it is easy to verify that assumptions V),VII) and condition that the functionα(t) is even in VI) don’t confine the general case if one examine the functionα1(t) = α(t)α(−t), where α(t) = (1 + α(t))(1 + t2). In [11, Theorems 1 and 2] ithas been proved that the monotony condition on α(t) in VI) doesn’t confine thegeneral case too.

It follows from VII) that

α(t)→∞, t→∞. (3.2)

Let β(t) = lnα(t), t ∈ R. Conditions III)-VII) and (3.2) lead to conclusion that

β(t) > 0, β(−t) = β(t), β(t)→∞, t→∞;

β(t1 + t2) ≤ β(t1) + β(t2), t1, t2 ∈ R (3.3)∫ ∞1

β(t)t2

dt <∞ (3.4)

1As shown in [8], for non-quasianalytic groups the condition ‖U(t)‖ ≥ 1 always holds, thereforein this paper the condition V) automatically takes place.

6 Ya. Grushka and S. Torba

Because of (3.3) there exists limit limt→∞β(t)t . And, by virtue of (3.4):

limt→∞

β(t)t

= 0. (3.5)

Also, using monotony of β(t), for t ∈ [k, k + 1], k ∈ N one gets

β(t)t2≥ β(k)

(k + 1)2≥ 1

4β(k)k2

,

thus using (3.4) it is easy to obtain that∞∑k=1

β(k)k2

<∞, (3.6)

moreover, all terms of series (3.6) are positive. From the convergence of series(3.6) it follows the existence of such sequence {Qn}∞n=1 ⊂ R that Qn > 1, Qn →∞, n→∞ and

∞∑k=1

β(k)k2

Qk = S <∞. (3.7)

We set

ak :=β(k)QkS k2

, k ∈ N.

The definition of ak and (3.7) result in equality∞∑k=1

ak = 1. (3.8)

We construct the sequence of functions, which, obviously, are entire for everyn ∈ N:

fn(z) :=n∏k=1

Pk(z), where Pk(z) =(

sin akz2

akz2

)2

, z ∈ C, n ∈ N.

Similarly to the proof of Denjoy-Carleman theorem [12, p.378] it can be concludedthat the sequence of (entire) functions fn(z) converges uniformly to the function

f(z) =∞∏k=1

(sin akz

2akz2

)2

, z ∈ C

in every disk {z ∈ C | |z| ≤ R}. Thus, by Weierstrass theorem, the function f(z)is entire.

It can be verified that 0 < ‖f‖L1(R) <∞. Thus let

Kα(z) :=1

‖f‖L1(R)

f(z), z ∈ C. (3.9)

Functions Kα(z) are generalizations of integral kernels (3.1). The following twolemmas state properties of these kernels.

Lemma 3.1 ([13]). Let α ∈ Q. Then the function Kα(z), constructed above, satisfies

Direct and Inverse Theorems in the Theory of Approximation ... 7

1) Kα(t) ≥ 0, t ∈ R;2)∫∞−∞Kα(t) dt = 1;

3) ∀r > 0 ∃cr = cr(α) > 0 ∀z ∈ C |Kα(rz)| ≤ cr er|Im z|

α(|z|)

Let α ∈ Q, and Kα : C 7→ C is the function constructed by the function α asabove. We set

Kα,r(z) := rKα(rz), z ∈ C, r ∈ (0,∞).

The lemma 3.1 ensures us that the function Kα,r has the following properties:

1) Kα,r(t) ≥ 0, t ∈ R;2)∫∞−∞Kα,r(t) dt = 1;

3) ∀z ∈ C |Kα,r(z)| ≤ rcr er|Im z|

α(|z|) ; r > 0.

Next lemma describes the behavior of derivatives of Kα,r(t).

Lemma 3.2 ([13]). For every r ∈ (0,∞) there exists a constant cr = cr(α) > 0such that for each n ∈ N the following inequality holds

|K(n)α,r(t)| ≤ cr

√2πnα

(nr

)α(|t|)

rn, t ∈ R.

Remark 3.3. If the function α(t) satisfies the conditions of lemma 3.1, but, more-over, has polynomial order of growth at infinity, i.e. ∃m ∈ N0, ∃M > 0:

α(t) ≤M(1 + |t|)2m, t ∈ R, (3.10)

another integral kernel may be used:

Kα(z) =1

Km

(sin z

2mz

2m

)2m

, Km =∫ ∞−∞

(sin x

2mx

2m

)2m

dx.

In much the same way as in lemmas 3.1 and 3.2 one can show that∣∣Kα(rz)∣∣ ≤ Cr er|Im z|

α(|z|), where Cr =

M

Km

(1 +

2mr

)2m

,

and ∣∣K(n)α,r(t)

∣∣ ≤ cr√2πnα(nr )α(|t|)

rn, where cr < 2rCr,

that is to say, defined in such a way integral kernel satisfies lemmas 3.1 and 3.2.

3.2. Spectral subspaces of non-quasianalytic operators

Main instrument for proving generalized Bernstein inequality is the theory of spec-tral subspaces of non-quasianalytic operator A, constructed in [8]. Recall thatspectral subspaces (denoted by L(∆)) are defined for all segments ∆ ⊂ R and arecharacterized by the following properties:

1) The operator A is defined on whole L(∆) and is bounded on it;2) L(∆) is invariant with respect to A;

8 Ya. Grushka and S. Torba

3) the spectrum of part A∆ of operator A, induced in L(∆), consists of inter-section of spectrum of A with the interior of segment ∆ and, perhaps, theendpoints of segment ∆. And at that, if the endpoint of segment ∆ does notbelong to the spectrum of A, it does not belong to the spectrum of A∆ either;

4) if there is some subspace L on which the operator A is defined everywhereand is bounded, and this subspace is invariant with respect to A, and atthe same time the spectrum of L-induced part of A is included in ∆, thenL ⊂ L(∆).Now we describe the construction of spectral subspaces and their main prop-

erties, and later state the relationship with the entire vectors of exponential type.Let θ(t) (−∞ < t <∞) is the entire function of order 1 with zeroes on the positiveimaginary ray:

θ(t) = C

∞∏k=1

(1− t

itk

), where 0 < t1 ≤ t2 ≤ . . . ,

∞∑k=1

1tk<∞, (3.11)

C is a constant. Define by E(∞)θ the class of entire functions φ(t) of finite type and

order 1 which satisfies for all m = 0, 1, . . . and for all a > 0 the condition

M(m,a)θ (φ) :=

∫ ∞−∞|tmθ(at)φ(t)| dt <∞. (3.12)

As shown in [8, Lemma 1.1.1], the Fourier transform of functions from E(∞)θ is

non-quasianalytic, that is the following property takes place:

Proposition 3.4. For any segment ∆ of real axis and for any open finite intervalI ⊃ ∆ there exists φ(t) ∈ E(∞)

θ such that its Fourier transform equals one in ∆and equals zero outside I.

Moreover, the class E(∞)θ is linear and is closed under convolutions and dif-

ferentiation.Next step is the construction of finite functions of operator A. For C0-group

with non-quasianalytic generator there exists [11] such entire function θ(t) of order1 with zeroes on the positive imaginary ray that

‖U(t)‖ ≤ |θ(t)| ∀t ∈ R.

Let’s consider arbitrary φ(t) ∈ E(∞)θ and construct linear operator

Pφ =∫ ∞−∞

φ(t)U(t) dt. (3.13)

The operator, defined by (3.13), is bounded due to (3.12). Next, consider arbitrarysegment ∆ of real axis and denote by E

(∞)θ (∆) the set of such functions φ(t) ∈

E(∞)θ that the Fourier transform φ(λ) = 1 in some interval containing ∆. Denote

by L(∆) the subspace of vectors x such that

Pφx = x (3.14)

Direct and Inverse Theorems in the Theory of Approximation ... 9

for all φ(t) ∈ E(∞)θ (∆).

Operators Pφ are useful to studying of vectors Anx and to proving of theBernstein inequality because of properties (3.13), (3.14) and property [8, p.445]

APφ = PφA = P−iφ′ , (3.15)

which allow to deal with derivatives of some entire functions instead of Banach-space operators and vectors.

The following theorem shows the close relationship between spectral sub-spaces and entire vectors of exponential type.

Theorem 3.5 ([14]). For all α > 0

Eα(A) ⊂ Ξα(A) = L([−α, α]),

moreover, Ξα(A) is the closed subspace of X.

4. Direct and Inverse theorems

In this section we establish generalizations of well-known Jackson’s inequality (di-rect theorem), Bernstein inequality and the inverse theorem.

4.1. Abstract Jackson-type inequality

Direct theorem can be proven in several ways, one of which is indirect — that is,we can show that there exists a vector or an approximation method giving goodestimate for the best approximation. But from the practical point of view moreinteresting is the method giving direct expression for the approximation, even ifthis approximation is not the best.

We show how to construct an approximation of a Banach-space vector byentire vectors of exponential type and state the main results (an analogue of theJackson’s inequality) for it.

Let the group {U(t) : t ∈ R} satisfies (1.1). Then it follows from [11, Theo-rems 1 and 2] that ∫ ∞

−∞

ln (MU (|t|))1 + t2

dt <∞. (4.1)

We fix arbitrary k ∈ N and set

α(t) :=(MU (|t|)

)k(1 + |t|)k+2, t ∈ R.

The function α is, obviously, even on R. Condition (4.1) and properties of functionMU (·) imply α ∈ Q, and, moreover,∫ ∞

−∞

((1 + |t|)MU (|t|)

)kα(t)

dt =∫ ∞−∞

dt

(1 + |t|)2= 2. (4.2)

Using lemma 3.1 (or remark 3.3 if α(t) ≤ M(1 + |t|)m) for the function α(t), weconstruct the family of kernels Kα,r.

10 Ya. Grushka and S. Torba

In what follows, we assume x ∈ X, r ∈ (0,∞) and n ∈ {1, . . . , k}. We define

xr,n :=∫ ∞−∞Kα,r(t)U(nt)x dt.

Let ν ∈ N0. It can be shown that xr,n ∈ C∞(A) =⋂ν∈N0

D(Aν) and

Aνxr,n =(−1)ν

nν

∫ ∞−∞K(ν)α,r(t)U(nt)x dt. (4.3)

Moreover, for the relation (4.3) we can get [13]

lim supν→∞

(‖Aνxr,n‖

)1/ν ≤ r

n.

The last inequality brings us to the conclusion that

xr,n ∈ E(A) and σ(xr,n, A) ≤ r

n, (4.4)

that is xr,n, as the entire vector of exponential type, constructed from vector x, isa candidate for constructing of the approximation.

For arbitrary x ∈ X we define

xr,k :=∫ ∞−∞Kα,r(t)(x+ (−1)k−1(U(t)− I)kx)dt =

=∫ ∞−∞Kα,r(t)

k∑n=1

(−1)n+1

(k

n

)U(nt)x dt (4.5)

Using definition (4.5) one can get

xr,k =k∑

n=1

(−1)n+1

(k

n

)∫ ∞−∞Kα,r(t)U(nt)xdt =

k∑n=1

(−1)n+1

(k

n

)xr,n.

Therefore, accordingly to (4.4),

xr,k ∈ E(A) and σ(xr,k, A) ≤ r.

Hence for an arbitrary x ∈ X we have

Er(x,A) = infy∈E(A) :σ(y,A)≤r

‖x− y‖ ≤ ‖x− xr,k‖ .

xr,k is the approximation of vector x. Remark that the rule by which we set thevector xr,k in correspondence to vector x is the linear operator

Lr,kx =∫ ∞−∞Kα,r(t)

k∑n=1

(−1)n+1

(n

k

)U(nt)x dt,

defined on all X, that is we constructed linear approximation method and gave thedirect expression for it.

Next we state the generalizations of Jackson’s inequality.

Direct and Inverse Theorems in the Theory of Approximation ... 11

Theorem 4.1 ([13]). Suppose that {U(t) : t ∈ R} satisfies condition (1.1). Thenfor every k ∈ N there exists a constant mk = mk(A) > 0, such that for all x ∈ Xthe following inequality holds

Er(x,A) ≤mk · ωk(

1r, x,A

), r ≥ 1. (4.6)

Remark 4.2. If, additionally, the group {U(t)} is bounded (MU (t) ≤ M <∞, t ∈R), then the assumption r ≥ 1 can be changed to r > 0.

Theorem 4.1 allows us to prove the analogue of classic Jackson’s inequalityfor m times differentiable functions.

Corollary 4.3 ([13]). Let x ∈ D(Am), m ∈ N0. Then for every k ∈ N0

Er(x,A) ≤mk+m

MU

(mr

)rm

ωk

(1r,Amx,A

), r ≥ 1, (4.7)

where constants mn (n ∈ N) are the same as in theorem 4.1.

By setting in corollary 4.3 k = 0 and taking into account that ω0 (·, Amx,A) ≡‖Amx‖, one can conclude the following inequality:

Corollary 4.4 ([13]). Let x ∈ D(Am), m ∈ N0. Then

Er(x,A) ≤ mm

rm(MU (1/r)

)m‖Amx‖ r ≥ 1, (4.8)

where the constants mn (n ∈ N) are the same as in theorem 4.1.

4.2. Generalized Bernstein inequality

One of the well-known inequalities in approximation theory is the Bernstein in-equality. If f(x) is an entire function of exponential type σ > 0, and

|f(x)| ≤M, −∞ < x <∞,then

|f ′(x)| ≤ σM, −∞ < x <∞. (4.9)This inequality was generalized for exponential type entire vectors in [14], butin contrast to the classical one, the generalized inequality ‖Anx‖ ≤ cnα

n‖x‖ hasthe constants cn in it. In this section we show that the constants cn grows slowlythan exponent and that in the general case of non-quasianalytic group U(t) it isimpossible to prove an analogue of Bernstein inequality with cn ≤ c uniformly forall n ∈ N.

Theorem 4.5 (Bernstein-type inequality). For every x ∈ E(A) with type, not ex-ceeding some α ≥ 1, the following inequality holds

‖Anx‖ ≤ cnαn‖x‖, (4.10)

where constants cn do not depend on x and α and are growing slowly than exponent,i.e. for any ε > 0 there exists such cε that cn ≤ cε(1 + ε)n for all positive integern.

12 Ya. Grushka and S. Torba

Proof. Let’s consider majorant θ(t) for the function ‖U(t)‖, constructed in [11].Remark that θ(t) is of the form (3.11). As it was done in (3.9) in section 2, by thefunction θ(t) one can construct the entire function K(t) of exponential type.

Let’s consider arbitrary ε > 0 and such function φα(t) that its Fourier trans-form equals 1 in [−α, α] and equals 0 outside

(− α(1 + 2ε), α(1 + 2ε)

). According

to [8, Lemma 1.1.1], one can use as φα(t) the function

φα(t) =K2(αε t) sin

(α(1 + ε)t

)πt

. (4.11)

Denote by

φ(t) :=K2(ε t) sin

((1 + ε)t

)πt

.

Then φα(t) = αφ(αt). As it follows from (3.13) and (3.15), it is enough to estimatethe quantity ∫ ∞

−∞|φ(n)α (t)θ(t)| dt.

to prove the theorem. For α ≥ 1 we have |θ(t)| ≤ |θ(αt)| and∫ ∞−∞|φ(n)α (t)θ(t)| dt ≤

∫ ∞−∞|φ(n)(αt)θ(αt)|αdt. (4.12)

The change of variables τ = α · t givesdnφ(αt)dtn

=dnφ(τ)dτn

· αn,

thus ∫ ∞−∞|φ(n)(αt)θ(αt)|αdt = αn ·

∫ ∞−∞|φ(n)(τ)θ(τ)| dτ.

It is easy to see that the last integral exists and does not depend on α. Let itequals cn > 0. Then

‖Anx‖ = ‖P(−i)nφ(n)x‖ ≤ cnαn‖x‖.

Let’s show that cn are growing slowly than exponent. Using the estimate [14, Proofof Theorem 1]

|φ(n)r,ε (t)| ≤ c(!)c2ε

√2πn(r + 2ε)n

∣∣∣∣θ(

nr+2ε

)θ(t)

∣∣∣∣2on derivatives of φr,ε(t) = K2(εt) sin rt

rt we obtain∣∣φ(n)(t)∣∣ ≤ c(!)c2ε

√2πn

π(1 + ε)(1 + 3ε)n

∣∣∣∣θ(n)θ(t)

∣∣∣∣2.Using [14, Proof of Theorem 1]

limn→∞

(∣∣∣θ2(nα

)∣∣∣)1/n

= 1, α ∈ R+,

and the arbitrary of ε > 0, from the last inequality we can get that cn grows slowlythen the exponent. �

Direct and Inverse Theorems in the Theory of Approximation ... 13

We see that the inequality (4.10) differs from direct analogue of classic one(4.9) by the constants cn, which equals 1 in the classic case. The question if theseconstants cn are the characteristic property of our generalization or only the im-perfection of proof, naturally arises. It is shown in [2, Theorem 1] that for boundedC0-group U(t) the direct analogue of Bernstein inequality (with cn ≡ 1) can beproved. But for non-quasianalytic operators the situation is different. In the sec-tion 5.2 we construct the example of operator A and vector x such that the type ofx equals some α > 1, but x 6∈ Eα(A), which means that there is no such constantc that

∀k ∈ N0

∥∥Akx∥∥ ≤ cαk,that is there exists a counterexample for any inequality of the form (4.10) withany bounded sequence cn.

As the consequence of theorem 4.5 we get the following estimate for operator∆kh:

Corollary 4.6 ([14]). Let x ∈ E(A) and σ(x) ≤ α, α ≥ 1. Then for all k ∈ N

‖∆khx‖ ≤ ck(hα)kMU (kh)‖x‖, (4.13)

where constants ck are the same as in the theorem 4.5 and the function MU (t) isdefined by (2.4).

4.3. Inverse theorem

Following results generalize classical Bernstein theorem (also known as inversetheorem).

Theorem 4.7 ([14]). Let ω(t) is the function of type of a module of continuity forwhich the following conditions are satisfied:

1. ω(t) is continuous and nondecreasing for t ∈ R+.2. ω(0) = 0.3. ∃c > 0 ∀t ∈ [0, 1] ω(2t) ≤ cω(t).4.∫ 1

0ω(t)t dt <∞.

If, for x ∈ X, there exist n ∈ N and m > 0 such that

Er(x,A) ≤ m

rnω

(1r

), r ≥ 1, (4.14)

then x ∈ D(An) and for every k ∈ N there exists a constant mk > 0 such that

ωk(t, Anx,A) ≤ mk

(tk∫ 1

t

ω(u)uk+1

du+∫ t

0

ω(u)u

du

), 0 < t ≤ 1/2. (4.15)

The following lemma is used for the proof of theorem.

Lemma 4.8 ([14]). Suppose that the function ω(t) satisfies conditions 1 – 3 oftheorem 4.7. If, for x ∈ X, there exists m > 0 such that

Er(x,A) ≤ mω(

1r

), r ≥ 1, (4.16)

14 Ya. Grushka and S. Torba

then, for every k ∈ N, there exists a constant ck > 0 such that

ωk(t, x,A) ≤ cktk∫ 1

k

ω(τ)τk+1

dτ, 0 < t ≤ 1/2. (4.17)

5. Examples of application of abstract direct and inverse theoremsin particular spaces

In this section we discuss two applications of presented theory. First is the ap-proximation of continuous functions by entire functions in the weighted Lp(R, µ)spaces with growing at the infinity weight (for example, L1(R, xn) spaces). Similarproblems studied in several papers (see the review [15]). Second is the approxima-tion of continuous functions by entire functions in the weighted Lp(R, µ) spaceswith decreasing at the infinity weight. Many papers, started from the works ofS.Bernstein [16, 17], concerned to this problem.

5.1. Direct and inverse theorems of approximation by exponential type entirefunctions in the space Lp(R, µp)

We consider a real-valued function µ(t) satisfying the following conditions:1) µ(t) ≥ 1, t ∈ R;2) µ(t) is even, monotonically non-decreasing when t > 0;3) µ(t) satisfies naturally occurring in many applications condition µ(t + s) ≤

µ(t) · µ(s), s, t ∈ R.4)∫∞−∞

lnµ(t)1+t2 dt <∞,

or alternatively, instead of 4), the equivalent condition holds:

4’)∑∞k=1

lnµ(k)k2 <∞.

Let’s consider several important classes of functions satisfying conditions 1)–4).

1. Constant function µ(t) ≡ 1, t ∈ R.2. Functions with polynomial order of growth at the infinity. It is easy to

check that for such functions following estimate holds:

∃k ∈ N, ∃M ≥ 1 µ(t) ≤M(1 + |t|)k, t ∈ R.

3. Functions of the form

µ(t) = e|t|β

, 0 < β < 1, t ∈ R.

4. µ(t) represented as a power series for t > 0. I.e.,

µ(t) =∞∑n=0

|t|n

mn,

where {mn}n∈N is the sequence of positive real numbers satisfying two conditions:• m0 = 1, m2

n ≤ mn−1 ·mn+1, n ∈ N;• ∀k, l ∈ N (k+l)!

mk+l≤ k!

mkl!ml

.

Direct and Inverse Theorems in the Theory of Approximation ... 15

The function µ(t), defined above, obviously satisfies conditions 1) and 2). Thecondition ∀k, l ∈ N (k+l)!

mk+l≤ k!

mkl!ml

impliesn∑k=0

tksn−kn!k!(n− k)!mn

≤n∑k=0

tksn−k

mkmn−k, (5.1)

and it is easy to see that condition 3) follows from inequality (5.1). Denjoy -Carleman theorem [12, p.376] asserts that the following conditions are equivalent:

a) µ(t) satisfies condition 4);

b)∑∞n=1

(1mn

)1/n

<∞;

c)∑∞n=1

mn−1mn

<∞.5. µ(t) as a module of an entire function with zeroes on the imaginary axis.

We consider

ω(t) = C

∞∏k=1

(1− t

itk

), t ∈ R,

where C ≥ 1, 0 < t1 ≤ t2 ≤ . . . ,∑∞k=1

1tk< ∞. We set µ(t) := |ω(t)|. Then µ(t)

satisfies conditions 1) – 3), and, as shown in [8], µ(t) satisfies condition 4) also.Let’s proceed to the description of spaces Lp(R, µp). Let the function µ(t)

satisfies conditions 1) – 4). We consider the space Lp(R, µp), 1 ≤ p ≤ ∞ offunctions x(s), s ∈ R, integrable in p-th degree with the weight µp:

‖x‖pLp(R,µp) =∫ ∞−∞|x(s)|pµp(s) ds.

Lp(R, µp) is the Banach space. We consider the differential operator

(Ax)(t) =dx

dt, D(A) = {x ∈ Lp(R, µp) ∩AC(R) : x′ ∈ Lp(R, µp)}.

The operator A generates the group of shifts {U(t)}t∈R in the space Lp(R, µp).This group isn’t bounded. As shown in [13],

‖U(t)‖Lp(R,µp) ≤ µ(|t|), t ∈ R.To obtain meaningful results of approximation by entire functions, we need

to establish how the space E(A) and the space of entire functions of exponentialtype are connected. Denote by Bσ the set of exponential functions of entire typeσ. We showed [14] that the following embedding holds

Ξσ(A) ⊂ Bσ ∩ Lp(R, µp). (5.2)

It was shown in [14] that for all functions µ(t), satisfying

µ(t) ≥ 1 +R|t|q

for some q > 1 − 1p , R > 0 and for all t > t0 ≥ 0, the reverse embedding holds.

In this paper we improve this result and show that Bσ ∩ Lp(R, µp) ⊂ Ξσ(A) forall weights µ(t), satisfying conditions 1)–4). Let f ∈ Bσ ∩ Lp(R, µp). Obviouslyf ∈ Lp(R). f is entire function, so (see [18, p.191]) it is bounded on the real axis.

16 Ya. Grushka and S. Torba

One can conclude that f ∈ S ′ (remind that S ′ is the Schwartz space of tempereddistributions, see [19]).

To prove that f ∈ Ξσ(A) by the use of theorem 3.5, let’s consider majorantθ(t) for the function µ(t), constructed as in the proof of theorem 4.5. We need toshow that for all φ ∈ E(∞)

θ ([−σ, σ])

f = Pφf =∫ ∞−∞

φ(t)U(t)fdt.

Note that it follows from the definition of class E(∞)θ that every such φ ∈ S.

Let’s consider φ ∈ E(∞)θ ([−σ, σ]). As φ ∈ S and f ∈ S ′, the convolution φ ∗ f is

well-defined [19, p.324]. But for φ1(t) = φ(−t)

(Pφ1f)(x) =∫ ∞−∞

φ1(t)U(t)f(x)dt =∫ ∞−∞

φ(t)f(x− t)dt = (φ ∗ f)(x).

The Fourier transform of φ ∗ f equals to

φ ∗ f = φ · f = f ,

because by the definition of E(∞)θ ([−σ, σ]) we have φ = 1 on [−σ, σ], and by the

Paley-Wiener theorem supp f ⊂ [−σ, σ]. Thus,

Pφ1f = f ∀φ1 ∈ E(∞)θ ([−σ, σ]),

(since φ(−t) also runs over all E(∞)θ ([−σ, σ]) so f ∈ L([−σ, σ]) and by the means

of theorem 3.5, f ∈ Ξσ(A).We have shown that Ξσ(A) = Bσ ∩ Lp(R, µp) for all weights µ(t), satisfying

conditions 1)–4). Note that ‖f−gσ‖Lp(R,µp) is defined only for those functions thatbelongs to Lp(R, µp) (because of ‖gσ‖Lp(R,µp) ≤ ‖f − gσ‖Lp(R,µp) + ‖f‖Lp(R,µp)),thus the best approximation by exponential type entire vectors is the same as thebest approximation by entire functions of exponential type.

By applying theorems 4.1, 4.5, 4.7 and corollary 4.3 we get several results forthe approximation theory in Lp(R, µp) spaces.

Corollary 5.1. For every k ∈ N there exists a constant mk(p, µ) > 0 such that forall f ∈ Lp(R, µp)

Er(f) ≤mk · ωk(

1r, f

), r ≥ 1.

Corollary 5.2. Let f ∈Wmp (R, µp), m ∈ N0. Then for all k ∈ N0

Er(f) ≤mk+m

µ(mr

)rm

ωk

(1r, f (m)

), r ≥ 1,

where constants mn (n ∈ N) are the same as in corollary 5.1.

Corollary 5.3. Let f ∈ Lp(R, µp)∩Bσ, σ ≥ 1. Then for all n ∈ N there exist suchconstants cn > 0, not depending on σ and growing slower than exponent, that

‖f (n)‖Lp(R,µp) ≤ cnσn‖f‖Lp(R,µp).

Direct and Inverse Theorems in the Theory of Approximation ... 17

Corollary 5.4. Let ω(t) be a function of type of a module of continuity for whichthe following conditions are satisfied:

1. ω(t) is continuous and nondecreasing for t ∈ R+.2. ω(0) = 0.3. ∃c > 0 ∀t ∈ [0, 1] ω(2t) ≤ cω(t).4.∫ 1

0ω(t)t dt <∞.

If, for f ∈ Lp(R, µp) there exist such n ∈ N and m > 0 that

Er(f) ≤ m

rnω

(1r

), r ≥ 1,

then f ∈Wnp (R, µp) and for every k ∈ N there exists a constant mk > 0 such that

ωk(t, f (n)) ≤ mk

(tk∫ 1

t

ω(u)uk+1

du+∫ t

0

ω(u)u

du

), 0 < t ≤ 1/2.

5.2. Exponential type entire functions in the space Lp(R, µ−p) and constants inBernstein inequality

We consider the same real-valued function µ(t) as in the previous subsection, butanother functional space. Consider the space Lp(R, µ−p) of functions x(s), s ∈ R,integrable in p-th degree with the weight µ−p:

‖x‖pLp(R,µ−p) =∫ ∞−∞|x(s)|pµ−p(s) ds.

Lp(R, µ−p) is the Banach space. We consider the same differential operator as inthe previous subsection

(Ax)(t) =dx

dt, D(A) = {x ∈ Lp(R, µ−p) ∩AC(R) : x′ ∈ Lp(R, µ−p)}.

The operator A generates the group of shifts {U(t)}t∈R in the space Lp(R, µ−p).This group isn’t bounded for unbounded µ(t). Let’s show this and that the groupis non-quasianalytic. Indeed, let’s consider for arbitrary t ≥ 1

x(s) =

{1, s ∈ [t, t+ 1],0, s 6∈ [t, t+ 1].

Obviously, x(s) ∈ Lp(R, µ−p). We have

‖U(t)x‖p =∫ ∞−∞|x(t+ s)|pµ−p(s) ds =

∫ 1

0

µ−p(s) ds = C > 0,

the last quantity does not depend on t. But

‖x‖p =∫ ∞−∞|x(s)|pµ−p(s) ds =

∫ t+1

t

µ−p(s) ds,

and from the monotony of µ(s) for s ∈ [0, 1] we get1

µ(t+ 1)≤ 1µ(t+ s)

≤ 1µ(t)

,

18 Ya. Grushka and S. Torba

so ‖x‖p ≤ 1µp(t) . For unbounded µ(t) this shows that

‖U(t)‖ ≥ ‖U(t)x‖‖x‖

≥ Cµ(t)→∞,

that is the group U(t) is unbounded.From the other hand, for every function x, because of the property 3),

‖U(t)x‖p =∫ ∞−∞

∣∣∣∣x(t+ s)µ(s)

∣∣∣∣p ds =∫ ∞−∞

∣∣∣∣x(t+ s) · µ(t)µ(s) · µ(t)

∣∣∣∣p ds ≤≤∫ ∞−∞

∣∣∣∣x(t+ s) · µ(t)µ(t+ s)

∣∣∣∣p ds =(µ(t)

)p‖x‖p,so ‖U(t)‖Lp(R,µ−p) ≤ µ(t), that is the group U(t) is non-quasianalytic.

Now we construct an example of vector such that the type of x equals someα > 1, but x 6∈ Eα(A). Consider µ(t) = (1 + |t|)3 and (for the simplicity)

x(t) = t · eikt, k ∈ N.

Then we can calculate

Anx(t) =(t · eikt

)(n) = n(ik)n−1eikt + (ik)nt · eikt.

Let

a =∫ ∞∞|eikt|pµ−p(t) dt, b =

∫ ∞∞|t · eikt|pµ−p(t) dt.

It is easy to see that 0 < a <∞ and 0 < b <∞, and from the property |y| − |z| ≤|y + z| ≤ |y|+ |z| of the absolute value we obtain

nkn−1a− knb ≤ ‖Anx‖Lp(R,µ−p) ≤ nkn−1a+ knb. (5.3)

It can be seen from the inequality (5.3) that there is no such constant c > 0 that‖Anx‖Lp(R,µ−p) ≤ c · kn for all n ∈ N, that is x 6∈ Ek(A), but, on the other hand,for any ε > 0 there exists such constant cε that ‖Anx‖Lp(R,µ−p) ≤ cε · (k+ ε)n, thatis x ∈ Ξk(A) and σ(x) = k.

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Yaroslav GrushkaInstitute of Mathematics of the National Academy of Sciences of Ukraine, Tereshchenkovskaya3, 01601 Kiev (Ukraine)e-mail: grushka@imath.kiev.ua

Sergiy TorbaInstitute of Mathematics of the National Academy of Sciences of Ukraine, Tereshchenkovskaya3, 01601 Kiev (Ukraine)e-mail: sergiy.torba@gmail.com